ITP001 Axioms: ITP103^7.ax
%------------------------------------------------------------------------------
% File : ITP103^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : tc.ax [Gau19]
% : HL4103^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 81 ( 20 unt; 40 typ; 0 def)
% Number of atoms : 194 ( 37 equ; 3 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 612 ( 3 ~; 6 |; 17 &; 563 @)
% ( 13 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 7 avg; 563 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 240 ( 240 >; 0 *; 0 +; 0 <<)
% Number of symbols : 41 ( 39 usr; 1 con; 0-7 aty)
% Number of variables : 177 ( 0 ^ 131 !; 5 ?; 177 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Efinite__map_2Efmap,type,
tyop_2Efinite__map_2Efmap: $tType > $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Erelation_2EEMPTY__REL,type,
c_2Erelation_2EEMPTY__REL:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Efinite__map_2EFAPPLY,type,
c_2Efinite__map_2EFAPPLY:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > A_27b ) ).
thf(c_2Efinite__map_2EFDOM,type,
c_2Efinite__map_2EFDOM:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > $o ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Etc_2EFMAP__TO__RELN,type,
c_2Etc_2EFMAP__TO__RELN:
!>[A_27a: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) > A_27a > A_27a > $o ) ).
thf(c_2Efinite__map_2EFUN__FMAP,type,
c_2Efinite__map_2EFUN__FMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).
thf(c_2Ecombin_2EI,type,
c_2Ecombin_2EI:
!>[A_27a: $tType] : ( A_27a > A_27a ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Elist_2ELIST__TO__SET,type,
c_2Elist_2ELIST__TO__SET:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Erelation_2EO,type,
c_2Erelation_2EO:
!>[A_27g: $tType,A_27h: $tType,A_27k: $tType] : ( ( A_27h > A_27k > $o ) > ( A_27g > A_27h > $o ) > A_27g > A_27k > $o ) ).
thf(c_2Erelation_2ERDOM,type,
c_2Erelation_2ERDOM:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27a > $o ) ).
thf(c_2Etc_2ERELN__TO__FMAP,type,
c_2Etc_2ERELN__TO__FMAP:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) ) ).
thf(c_2Erelation_2ERTC,type,
c_2Erelation_2ERTC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2ERUNION,type,
c_2Erelation_2ERUNION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Erelation_2ETC,type,
c_2Erelation_2ETC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Etc_2ETC__ITER,type,
c_2Etc_2ETC__ITER:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) ) ).
thf(c_2Etc_2ETC__MOD,type,
c_2Etc_2ETC__MOD:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Etc_2E_5E_7C,type,
c_2Etc_2E_5E_7C:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Etc_2E_5E_7C_5E,type,
c_2Etc_2E_5E_7C_5E:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Efinite__map_2Eo__f,type,
c_2Efinite__map_2Eo__f:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ).
thf(c_2Etc_2EsubTC,type,
c_2Etc_2EsubTC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Etc_2E_7C_5E,type,
c_2Etc_2E_7C_5E:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Etc_2EDRESTR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a,V3b: A_27a] :
( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ V1s @ V2a @ V3b )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s )
& ( V0R @ V2a @ V3b ) ) ) ).
thf(thm_2Etc_2ERRESTR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a,V3b: A_27a] :
( ( c_2Etc_2E_7C_5E @ A_27a @ V0R @ V1s @ V2a @ V3b )
<=> ( ( c_2Ebool_2EIN @ A_27a @ V3b @ V1s )
& ( V0R @ V2a @ V3b ) ) ) ).
thf(thm_2Etc_2EBRESTR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s )
= ( c_2Etc_2E_7C_5E @ A_27a @ ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ V1s ) @ V1s ) ) ).
thf(thm_2Etc_2EsubTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V2x @ V3y )
<=> ( ( V0R @ V2x @ V3y )
| ? [V4a: A_27a,V5b: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V4a @ V5b )
& ( c_2Ebool_2EIN @ A_27a @ V4a @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V5b @ V1s )
& ( V0R @ V2x @ V4a )
& ( V0R @ V5b @ V3y ) ) ) ) ).
thf(thm_2Etc_2EFMAP__TO__RELN,axiom,
! [A_27a: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ),V1x: A_27a] :
( ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0f @ V1x )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V0f ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( A_27a > $o ) @ V0f @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Etc_2ERELN__TO__FMAP,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Etc_2ERELN__TO__FMAP @ A_27a @ V0R )
= ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ ( A_27a > $o ) @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ) ).
thf(thm_2Etc_2ETC__MOD,axiom,
! [A_27a: $tType,V0x: A_27a,V1rx: A_27a > $o,V2ra: A_27a > $o] :
( ( c_2Etc_2ETC__MOD @ A_27a @ V0x @ V1rx @ V2ra )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V2ra ) @ ( c_2Epred__set_2EUNION @ A_27a @ V2ra @ V1rx ) @ V2ra ) ) ).
thf(thm_2Etc_2ETC__ITER,axiom,
! [A_27a: $tType] :
( ! [V0r: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
( ( c_2Etc_2ETC__ITER @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) @ V0r )
= V0r )
& ! [V1x: A_27a,V2d: tyop_2Elist_2Elist @ A_27a,V3r: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
( ( c_2Etc_2ETC__ITER @ A_27a @ ( c_2Elist_2ECONS @ A_27a @ V1x @ V2d ) @ V3r )
= ( c_2Etc_2ETC__ITER @ A_27a @ V2d @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( A_27a > $o ) @ ( A_27a > $o ) @ ( c_2Etc_2ETC__MOD @ A_27a @ V1x @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( A_27a > $o ) @ V3r @ V1x ) ) @ V3r ) ) ) ) ).
thf(thm_2Etc_2EDRESTR__IN,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a] :
( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ V1s @ V2a )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) @ ( V0R @ V2a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Etc_2EDRESTR__EMPTY,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ).
thf(thm_2Etc_2EDRESTR__RDOM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
= V0R ) ).
thf(thm_2Etc_2EREMPTY__RRESTR,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Etc_2E_7C_5E @ A_27a @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) @ V0s )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ).
thf(thm_2Etc_2EO__REMPTY__O,axiom,
! [A_27a: $tType] :
( ! [V0R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) )
& ! [V1R: A_27a > A_27a > $o] :
( ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) @ V1R )
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ).
thf(thm_2Etc_2EsubTC__thm,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s )
= ( c_2Erelation_2ERUNION @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ ( c_2Etc_2E_5E_7C @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) ) @ V1s ) @ V0R ) ) ) ) ).
thf(thm_2Etc_2EsubTC__EMPTY,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= V0R ) ).
thf(thm_2Etc_2ERTC__INSERT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a,V3w: A_27a,V4z: A_27a] :
( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2a @ V1s ) ) @ V3w @ V4z )
<=> ( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V3w @ V4z )
| ( ( ( V2a = V3w )
| ? [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ V1s )
& ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V3w @ V5x )
& ( V0R @ V5x @ V2a ) ) )
& ( ( V2a = V4z )
| ? [V6y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V6y @ V1s )
& ( V0R @ V2a @ V6y )
& ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V6y @ V4z ) ) ) ) ) ) ).
thf(thm_2Etc_2EsubTC__INSERT,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2q: A_27a,V3x: A_27a,V4y: A_27a] :
( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2q @ V1s ) @ V3x @ V4y )
<=> ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3x @ V4y )
| ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3x @ V2q )
& ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V2q @ V4y ) ) ) ) ).
thf(thm_2Etc_2EsubTC__RDOM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
= ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ).
thf(thm_2Etc_2EsubTC__INSERT__COR,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3a: A_27a] :
( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1s ) @ V3a )
= ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3a ) ) @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3a ) @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V2x ) ) @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3a ) ) ) ).
thf(thm_2Etc_2ERDOM__SUBSET__FDOM,axiom,
! [A_27a: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0f ) ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V0f ) ) ).
thf(thm_2Etc_2EFINITE__RDOM,axiom,
! [A_27a: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0f ) ) ) ).
thf(thm_2Etc_2EFDOM__RDOM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
=> ( ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a @ V0R ) )
= ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ) ).
thf(thm_2Etc_2ERELN__TO__FMAP__TO__RELN__ID,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
=> ( ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a @ V0R ) )
= V0R ) ) ).
thf(thm_2Etc_2ERDOM__subTC,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s ) )
= ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ).
thf(thm_2Etc_2ENOT__IN__RDOM,axiom,
! [A_27a: $tType,V0Q: A_27a > A_27a > $o,V1x: A_27a] :
( ( ( V0Q @ V1x )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0Q ) ) ) ) ).
thf(thm_2Etc_2ETC__MOD__EMPTY__ID,axiom,
! [A_27a: $tType,V0x: A_27a,V1ra: A_27a > $o] :
( ( c_2Etc_2ETC__MOD @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
= ( c_2Ecombin_2EI @ ( A_27a > $o ) ) ) ).
thf(thm_2Etc_2EI__o__f,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
( ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27b @ ( c_2Ecombin_2EI @ A_27b ) @ V0f )
= V0f ) ).
thf(thm_2Etc_2EsubTC__MAX__RDOM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a] :
( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) )
=> ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1s ) )
= ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s ) ) ) ).
thf(thm_2Etc_2EsubTC__SUPERSET__RDOM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
=> ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) @ V1s ) )
= ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ) ) ).
thf(thm_2Etc_2EsubTC__FDOM,axiom,
! [A_27a: $tType,V0g: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ),V1R: A_27a > A_27a > $o] :
( ( ( c_2Etc_2EsubTC @ A_27a @ V1R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V1R ) )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0g ) )
=> ( ( c_2Etc_2EsubTC @ A_27a @ V1R @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V0g ) )
= ( c_2Etc_2EsubTC @ A_27a @ V1R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V1R ) ) ) ) ).
thf(thm_2Etc_2ESUBSET__FDOM__LEM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
( ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V2f ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V2f ) ) ) ).
thf(thm_2Etc_2EsubTC__FDOM__RDOM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
( ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V1f ) )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V1f ) )
=> ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V1f ) ) ) ).
thf(thm_2Etc_2ETC__MOD__LEM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V3f ) )
& ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V3f ) ) )
=> ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1s ) )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( A_27a > $o ) @ ( A_27a > $o ) @ ( c_2Etc_2ETC__MOD @ A_27a @ V2x @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( A_27a > $o ) @ V3f @ V2x ) ) @ V3f ) ) ) ) ).
thf(thm_2Etc_2ETC__ITER__THM,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1d: tyop_2Elist_2Elist @ A_27a,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ),V3s: A_27a > $o] :
( ( ( ( c_2Epred__set_2EUNION @ A_27a @ V3s @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1d ) )
= ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V2f ) )
& ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V3s )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V2f ) ) )
=> ( ( c_2Erelation_2ETC @ A_27a @ V0R )
= ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ ( c_2Etc_2ETC__ITER @ A_27a @ V1d @ V2f ) ) ) ) ).
%------------------------------------------------------------------------------